Notes on the Loop Equation in Loop Space

TL;DR

This paper reformulates the loop equation in QCD as a functional Laplace equation, discretizes loop space, and uses path integrals to iteratively solve it, reproducing Feynman diagrams up to order (g^2N)^2.

Contribution

It introduces a novel discretization and path integral approach to solve the loop equation in QCD, connecting it with Feynman diagrams.

Findings

Reformulation of the loop equation as a functional Laplace equation.

Representation of Green's function as a path integral of the Euclidean harmonic oscillator.

Successful reproduction of Feynman diagrams up to order (g^2N)^2.

Abstract

The loop equation satisfied by Wilson's loops in QCD is reformulated as a functional Laplace equation. Discretizing the loop space by polygons, Green's function of the functional Laplacian is represented as a path integral of the Euclidean harmonic oscillator and is applied for an iterative solution of the equation. It is shown how the usual Feynman's diagrams are reproduced through order $(g^2N)^2$ including the one with the three-gluon vertex.